{"problem":{"name":"Two LIS Sum","description":{"content":"For a sequence $P = (P_1, \\ldots, P_N)$, let $\\mathrm{LIS}(P)$ denote the length of a longest increasing subsequence. You are given permutations $A = (A_1, \\ldots, A_N)$ and $B = (B_1, \\ldots, B_N)$ o","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc149_b"},"statements":[{"statement_type":"Markdown","content":"For a sequence $P = (P_1, \\ldots, P_N)$, let $\\mathrm{LIS}(P)$ denote the length of a longest increasing subsequence.\nYou are given permutations $A = (A_1, \\ldots, A_N)$ and $B = (B_1, \\ldots, B_N)$ of integers from $1$ through $N$. You may perform the following operation on these sequences any number of times (possibly zero).\n\n*   Choose an integer $i$ such that $1\\leq i\\leq N-1$. Swap $A_i$ and $A_{i+1}$, and swap $B_i$ and $B_{i+1}$.\n\nFind the maximum possible final value of $\\mathrm{LIS}(A) + \\mathrm{LIS}(B)$.\nWhat is a longest increasing subsequence?A subsequence of a sequence is a sequence obtained by removing zero or more elements from the original sequence and then concatenating the remaining elements without changing the order. For instance, $(10,30)$ is a subsequence of $(10,20,30)$, but $(20,10)$ is not a subsequence of $(10,20,30)$.\nA longest increasing subsequence of a sequence is a subsequence of that sequence with the greatest length among its subsequences that are strictly increasing.\n\n## Constraints\n\n*   $2\\leq N\\leq 3\\times 10^5$\n*   $1\\leq A_i\\leq N$\n*   $1\\leq B_i\\leq N$\n*   $A_i\\neq A_j$ and $B_i\\neq B_j$, if $i\\neq j$.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$\n$A_1$ $\\ldots$ $A_N$\n$B_1$ $\\ldots$ $B_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc149_b","tags":[],"sample_group":[["5\n5 2 1 4 3\n3 1 2 5 4","8\n\nHere is one way to achieve $\\mathrm{LIS}(A) + \\mathrm{LIS}(B) = 8$.\n\n*   Do the operation with $i = 2$. Now, $A = (5,1,2,4,3)$, $B = (3,2,1,5,4)$.\n*   Do the operation with $i = 1$. Now, $A = (1,5,2,4,3)$, $B = (2,3,1,5,4)$.\n*   Do the operation with $i = 4$. Now, $A = (1,5,2,3,4)$, $B = (2,3,1,4,5)$.\n\nHere, $A$ has a longest increasing subsequence $(1,2,3,4)$, so $\\mathrm{LIS}(A)=4$, and $B$ has a longest increasing subsequence $(2,3,4,5)$, so $\\mathrm{LIS}(B)=4$."],["5\n1 2 3 4 5\n1 2 3 4 5","10\n\nYou can decide not to perform the operation at all to achieve $\\mathrm{LIS}(A) + \\mathrm{LIS}(B) = 10$."]],"created_at":"2026-03-03 11:01:14"}}